Reciprocity (electromagnetism) - Reciprocity For Electrical Networks

Reciprocity For Electrical Networks

Above, Lorentz reciprocity was phrased in terms of an externally applied current source and the resulting field. Often, especially for electrical networks, one instead prefers to think of an externally applied voltage and the resulting currents. The Lorentz reciprocity theorem describes this case as well, assuming ohmic materials (i.e. currents that respond linearly to the applied field) with a 3×3 conductivity matrix σ that is required to be symmetric, which is implied by the other conditions below. In order to properly describe this situation, one must carefully distinguish between the externally applied fields (from the driving voltages) and the total fields that result (King, 1963).

More specifically, the above only consisted of external "source" terms introduced into Maxwell's equations. We now denote this by to distinguish it from the total current produced by both the external source and by the resulting electric fields in the materials. If this external current is in a material with a conductivity σ, then it corresponds to an externally applied electric field where, by definition of σ:

Moreover, the electric field above only consisted of the response to this current, and did not include the "external" field . Therefore, we now denote the field from before as, where the total field is given by .

Now, the equation on the left-hand side of the Lorentz reciprocity theorem can be rewritten by moving the σ from the external current term to the response field terms, and also adding and subtracting a term, to obtain the external field multiplied by the total current :

For the limit of thin wires, this gives the product of the externally applied voltage (1) multiplied by the resulting total current (2) and vice versa. In particular, the Rayleigh-Carson reciprocity theorem becomes a simple summation:

where V and I denote the (complex) amplitudes of the AC applied voltages and the resulting currents, respectively, in a set of circuit elements (indexed by n) for two possible sets of voltages and .

Most commonly, this is simplified further to the case where each system has a single voltage source V, at and . Then the theorem becomes simply : the current at position (1) from a voltage at (2) is identical to the current at (2) from the same voltage at (1).